In the cluster expansion process, we consider the C\(_2\) T\(_4\)  alloy
(see Fig. REF a)
with the top sites T occupied by H atoms or
vacancies. Our calculation shows that there are four important
pair interactions as shown in Fig. REF a, and the three-body
interactions are negligible. We can see that the interaction between
two H atoms adsorbed on the same C atom is extremely repulsive
(0.340 eV). This is understandable because the five-fold coordinated
carbon is not stable.
In contrast, we find that the interaction parameter
between two H atoms adsorbed on different sides of the two adjacent C
atoms is largely negative (\(-0.276\)  eV). The efficient strain
relaxation and the absence of dangling \(\pi \)  bonds in
this four-fold coordinated carbon
configuration account for the stability.
Using the cluster expansion Hamiltonian, we can easily obtain the
energy of a given alloy configuration and thus the ground state
structure of the partially hydrogenated graphene for a given
supercell.
It is clear that the number of H atoms (n[H]) should not exceed that
of C atoms (n[C]) because otherwise some C atoms will bind with more than
one H atoms.
When n[H]/n[C] \(= 1\) ,
the ground state structure is graphane (see
Fig. REF b), as found by Elias et al.. [1]}
For n[H]/n[C] \(= 0.5\) , the lowest energy
structure among all possible configurations with no more than 4 carbon
atoms per supercell is shown in Fig. REF c.
The adsorbed H atoms adsorb on a 1D zigzag carbon chain such that
each H atom has two neighbor H atoms adsorbed on the opposite side of
neighboring C atoms. This is consistent with the fact that the hydrogenation
of neighbor C atoms from opposite sides is energetically preferred.
However, we find that the above structure is not the global ground state structure
with n[H]/n[C] \(=0.5\) . For example, the lowest energy
structure (Fig. REF d) among all possible configurations with no
more than 8 carbon atoms per supercell has a lower energy by 44 meV/C
than does the structure shown in Fig. REF c.
This is so because the number of carbon-carbon \(p_z\) -\(p_z\)  \(\pi \)  bonds is
increased from 0.5/C (Fig. REF c ) to 0.625/C (Fig. REF d).
It is expected that the global ground state
of the partially hydrogenated graphene displays macroscopic phase
separation between graphene and graphane regions.
To confirm this point, we consider a large supercell
(\(8\times 8\) ) with
24 adsorbed H atoms. Using the Monte Carlo annealing technique, we find all the H
atoms form a close packed cluster in the ground state configuration
(Fig. REF e).
This shows that, in the global ground state of a partially
hydrogenated graphene, the phase separation into graphene and graphane
parts would take place.
